Archive for July, 2015

A Short History of Quantum Reference Frames, Part 2

Posted in Uncategorized on July 18, 2015 by quantummoxie

Yesterday, I began outlining the history of quantum reference frames beginning with Marco Toller’s paper in 1977. What we saw was two, somewhat separate lines of development. The first was a direct line from Toller through Rovelli and back again to Toller, in which a generalized (and distinctly operational) notion of a reference frame is developed that covers both quantum and classical systems. The second was a line beginning with Holevo and leading up through Peres and Scudo, in which a protocol is developed for sending information about a Cartesian coordinate system via a quantum channel.

The question I have is, where do the lines converge? Specifically, when on the informational side do people start referring to reference frames in the more general sense? Pinpointing an exact instance is difficult, but an early paper by Bartlett, Rudolph, and Spekkens (hereafter referred to as BRS) in 2003 seems to suggest that Toller’s idea had made its way into the quantum information community by that time. The first few sentences of their paper read:

Quantum physics allows for powerful new communication tasks that are not possible classically, such as secure communication and entanglement-enhanced classical communication. In investigations of these and other communication tasks, considerable effort has been devoted to identifying the physical resources that are required for their implementation. It is generally presumed, at least implicitly, that a shared reference frame (SRF) between the communicating parties is such a resource, with the precise nature of the reference frame being dictated by the particular physical systems involved.

The emphasis on the word ‘physical’ in the second sentence is theirs. The emphasis in the last sentence is mine and meant to show that Toller’s idea had gained some traction in the QI community by then. In a meaty review article published in 2007, BRS offered a slightly different take on Toller’s notion of a reference frame (and notably still did not cite his work which makes me wonder if they were and still are unaware of it, or if they disagreed with his definition to some extent). They refer back to the work by Peres and Scudo and define two types of information: fungible and nonfungible (referred to by Peres and Scudo as ‘speakable’ and ‘unspeakable’ respectively). Fungible information is typically classical and refers to information for which the means of encoding does not matter. The example they give is that Shannon’s coding theorems don’t care one way or the other how the 0’s and 1’s are encoded (e.g. by magnetic tape or by voltages or some other manner). So this is fungible information. If, on the other hand, the encoding of information does make a difference to the information being encoded, then that information is said to be nonfungible (unspeakable). BRS then note that

[w]e refer to the systems with respect to which unspeakable/nonfungible information is defined, clocks, gyroscopes, metre sticks and so forth, as reference frames.

The emphasis is theirs. How does this compare to Toller’s definition of a frame of reference as some material object that is of the same nature as the objects that form the system under investigation, as well as the instruments used to measure that system? It seems to me that the BRS definition is merely a more concise statement of Toller’s definition which does not account for the differences between fungible and nonfungible information. So I think we’re still talking about the same thing here.

Superficially, it is fairly easy to see that many quantum ideas are natural generalizations of classical ideas. For example, in his terrific (and freely available!) undergraduate course materials on the mathematics of theoretical physics, Karl Svozil defines a reference frame (i.e. a “coordinate system”) as a linear basis. This is, arguably, a non-operational alternative to Toller’s definition, but I think the two can essentially be made equivalent. In fact he launches into a detailed discussion of the motivation behind defining the Cartesian reference frame which he refers to as “the Cartesian basis.” So the Cartesian frame of reference is just a special case of the broader idea of a degree of freedom as I pointed out in yesterday’s post. The point of all this is, even engineers will recognize that the even the classical notion of a reference frame is more than purely Cartesian (aerospace engineers, for example, work in systems with six degrees of freedom – either the three spatial coordinates plus the Euler angles [referred to by flight dynamics people as pitch, roll, and yaw], or the three spatial coordinates plus the corresponding momenta in each of those directions, i.e. phase space).

At any rate, one of the key results given in the BRS review has to do with the relation between quantum systems and classical systems. It is built on the notion of a superselection rule (SSR) which I briefly mentioned in yesterday’s post. SSRs were introduced by Wick, Wightman, and Wigner in 1952. In essence, they are rules that place limits on certain types of measurements, but are perhaps better understood as rules that prohibit superpositions of certain coherent states. What BRS showed in their review was that SSRs are formally equivalent to the lack of a shared classical reference frame (in the general sense of the term). So for instance, Wick, Wightman, and Wigner suggested that an SSR existed for states of opposite charge, i.e. one would never see a superposition of positive and negative charge. It makes perfect sense that this is equivalent to the lack of a classical reference frame for charge. In other words, we will never find a metal plate that is simultaneously positively charged and negatively charged. It is always one or the other or neutral. This might pose a problem for two parties (Alice and Bob again) who might, for some bizarre reason, have independently designed communication devices that rely on different charges. In a highly contrived example, suppose Bob lives on a planet that developed computers that run on positive charge somehow (maybe ionized atoms, for example), whereas Alice lives here on earth where our computers run on negative charge. If both use voltage readings to determine 0’s and 1’s in their binary codes, then they will get completely opposite results if they try to interpret each others’ machines. This is a completely classical issue.

On the other hand, at the quantum level, states exist that are a superposition of opposite charges despite the SSR (the first suggestion of this for charge was by Aharonov and Susskind in 1967 which is the paper that essentially launched our work on CPT symmetry). So while Alice and Bob may lack a classical reference frame for their computers, a common quantum reference frame can be established by using these superposition states as building blocks. Understanding this point really requires understanding precisely what a reference frame is as well as how it is used in communicating information between two parties. In other words, it requires understanding the difference between fungible and nonfungible information. It’s just another example of how an information theoretical view can shed light on some of the deepest (and seemingly uncontroversial) problems in physics.

A Short History of Quantum Reference Frames, Part 1

Posted in Uncategorized on July 17, 2015 by quantummoxie

I was recently at the Relativistic Quantum Information – North (RQI-N) meeting at Dartmouth College where I presented some of the work I have been doing on quantum reference frames and how to use them to overcome certain superselection rules (SSRs), specifically the SSR associated with CPT symmetry. I received a lot of terrific comments and numerous discussions have been spawned from the presentation. In particular, I seem to have introduced a number of people to the concept of a quantum reference frame for the first time. In fact I’m meeting back up with a few folks next week to discuss some of these issues. In preparing for next week’s discussions, I decided to do a little historical research to find out where the concept originated and how it got to its current form. I thought a blog post on the topic might be somewhat useful to some people out there and it would also provide me with a permanent record of some of the background papers in the field.

So what is a quantum reference frame? The idea appears to have first been proposed by Marco Toller in 1977 in a paper entitled “An operational analysis of the space-time structure” that appeared in Il Nuovo Cimento B. Here I quote the abstract:

We discuss the concepts related to space-time in a quantum-relativistic theory by means of the analysis of the physical procedures used to construct a new frame of reference starting from a pre-existent frame (transformation procedures). The physical objects which form a frame of reference are allowed to interact with the other physical objects and follow the laws of quantum physics. We suggest that there are conceptual limitations which do not permit the exact realization of a transformation of the Poincaré group by means of physical procedures. We remark also that the operations performed in order to construct a frame of reference perturb the surrounding physical objects and are influenced by them. We propose some general theoretical schemes which take these facts into account and permit the separation of the geometrical effects of a transformation procedure from the physical ones. Finally we find the conditions which permit the construction of a Poincaré-invariant theory of the usual kind by means of the introduction of some ideal concepts which have no direct operational meaning.

In other words, the most general definition of a ‘frame of reference’ is as some material object that is of the same nature as the objects that form the system under investigation as well as the measuring instruments themselves (Bohr’s classical-quantum contrast not withstanding). This idea was further developed by Aharonov and Kaufherr in 1984 in which they extended the principle of equivalence to quantum reference frames, and in a pair of articles written in 1991 by Carlo Rovelli (see here and here) which appear to have played some role in inspiring his relational interpretation of quantum mechanics. In this way, these ideas bear a striking resemblance to work attempted by Eddington in the 1930s and early 1940s (a topic I will leave for another blog post, but that served as the core topic of my long-forgotten PhD thesis).

Anyway, these ideas are clearly operational (Toller even uses the term in his original paper). They were, however, not necessarily informational, at least initially. However, in his 1982 book Probabilistic and Statistical Aspects of Quantum Theory, Alexander Holevo (who was just announced as the 2016 winner of the Claude E. Shannon award by the IEEE Information Theory Society) addressed the following question: can a system of N elementary spins (i.e. qubits, which weren’t yet named in 1982) be used to communicate, in a single transmission, the orientation of three mutually orthogonal unit vectors, i.e. a Cartesian reference frame? Holevo concluded that if the system had a well-defined total spin angular momentum J then, at best, only one of the three vectors could be communicated. A way around this limitation was found nearly two decades later by Bagan, Baig, and Muñoz-Tapia and, around the same time, Peres and Scudo found a way to do it with a single hydrogen atom. The idea was to allow two distant parties (i.e. our old friends Alice and Bob) to establish a common Cartesian reference frame simply using a quantum channel. Thus these papers, while informational in their focus, used the less general definition of a reference frame as a Cartesian coordinate system. In fact it is not entirely clear that any of these authors (or others working on similar ideas – see the previously mentioned paper by Bagan, et. al. for additional references) was aware of the more general definition of the reference frame originally proposed by Toller.

One of the key ideas in the early information-related papers was that the Cartesian frame, i.e. the concept of a spatial direction, could be encoded in a particle’s spin state. Somewhere along the line (it’s not quite clear to me yet exactly when) someone put these two ideas together and the more general concept of a quantum reference frame was born. It appears that somewhere around 2002 or 2003 someone realized that a spatial direction is an example of a degree of freedom. Of course even classical physicists – even many engineers – know that there are more general and abstract spaces that have more than three degrees of freedom (e.g. phase space). For decomposable systems, a distinction can be made between what might be called ‘collective’ degrees of freedom, i.e. those between a system and something external to it, and ‘relative’ degrees of freedom, i.e. those between the systems constituent parts. Several authors (including John Preskill, who was at RQI-N) recognized that encoding information into the collective degrees of freedom posed a number of problems. Beginning, to the best of my knowledge, in 1997 with a paper by Zanardi and Rasetti, encoding information into the relative degrees of freedom of a system was shown to be more optimal in some situations. Hopefully, you can see where this is headed. The relational degrees of freedom harken back to the general frame of reference á là Rovelli and his relational interpretation of QM. For example, take a look at this early paper by Bartlett, Rudolph, and Spekkens. The first few paragraphs offer a fairly nice summary of some of the work that had just recently come out on relative quantum information, though the paper itself still primarily deals with something spatial.

As early as 1996, Toller himself recognized that limitations in representations of the Poincaré group necessitated taking “internal” degrees of freedom into account when working with quantum reference frames. An example of such an internal degree of freedom is electrical charge. In fact, in our first paper in PRL, we introduce a new quantum number that represents all of the universally conserved internal quantum degrees of freedom (which happen to only be electrical charge and the difference between baryon number and lepton number), though we were unaware of Toller’s paper at that point (in fact I was unaware of it until I started working on this blog post). It may well be, in fact, that we are the first to have considered internal degrees of freedom in such a manner.

At any rate, in Part 2 of this short history, I will attempt to nail down exactly who first suggested using a generalized reference frame in the manner of Toller in an information communication scheme. I will then discuss the relation to SSRs which play a vitally important role in this story.